Well I believe its put as 'more than a random' graph but I'm not
sure what that value is. Sounds like another statistical distribution.
sheep
On 4 Sep 2007, at 21:45, Alan Penn wrote:
> Can anyone answer the question of how many three/four/(any small
> number)
> cycles are needed out of a whole graph to make it a small world..
> or is this
> not defined?
>
> Alan
>>
>> Lucas,
>>
>> I'm not sure I understand your objection to either the original
>> clustering coefficient or Jiang and Claramunt's 2-step version. I
>> don't
>> even think that Jiang and Claramunt do "mix" cycle lengths in their
>> measure -- the measure as I understand it is equivalent to looking at
>> length-4 cycles, and almost identical to Calderelli et al's measure
>> (their approach simply mirrors Watts and Strogatz's original idea of
>> interconnectivity rather than considering the problem as cycles).
>>
>> I do like the point about simply using the number of cycles though.
>> Musing upon it a bit further though, isn't any highly clustered graph
>> (lots of cycles) likely to be a small world?
>>
>> Alasdair
>>
>> Lucas Figueiredo wrote:
>>> I prefer the Caldarelli and colleagues's definition:
>>> Caldarelli, G., R. Pastor-Satorras, et al. (2004). "Structure of
>>> cycles and local ordering in complex networks." Eur. Phys. J. B 38:
>>> 183-186.
>>>
>>> It extracts specifically cycles of k steps. 'K-clustering' mixes all
>>> cycles and probably is ill-defined as the original clustering
>>> coefficient, which is degree biased.
>>>
>>> Of course, there is the original one: 'axial ringness' (Hillier and
>>> Hanson, 1984).
>>>
>>> Best Regards,
>>> Lucas Figueiredo
>>>
>>> On 04/09/07, Alasdair Turner <[log in to unmask]> wrote:
>>>> Of course, there's always Bin Jiang.
>>>>
>>>> Here's the paper where he and Christophe Claramunt introduce the
>>>> k-clustering coefficient.
>>>>
>>>> Jiang B, Claramunt C, 2004, "Topological analysis of urban street
>>>> networks" Environment and Planning B: Planning and Design 31(1)
>>>> 151 -
>> 162
>>>>
>>>>
>>>> Lucas Figueiredo wrote:
>>>>> On 04/09/07, S. N.C. Dalton <[log in to unmask]> wrote:
>>>>>> Watts amd Strpgatz used clustering coefficient of a graph to
>>>>>> determine if a
>>>>>> graph is small world or not.
>>>>> Their definition is almost (already) 10 years old and it is
>>>>> restrictive. It is common now in the literature to check squares
>>>>> (cycles of 4 steps) instead of triangles. The important is the
>>>>> idea
>>>>> that elements are clustered (either in triangles, squares or even
>>>>> trees!) at local level but still have 'shortcuts' that connect
>>>>> them to
>>>>> the rest of the system in few steps.
>>>>>
>>>>> Local streets are definitively clustered, or at least this is
>>>>> the way
>>>>> I see it. Do not get emotionally attached to these
>>>>> 'definitions' of
>>>>> things by 'might scholars'.
>>>>>
>>>>> Challenge them! Contradict them! Innovate!
>>>>>
>>>>> Best Regards,
>>>>> Lucas Figueiredo
>>>>>
>>>>>> The only time you get a clustering coefficient bigger than
>>>>>> zero is
>>>>>> when 3 or more axial lines
>>>>>> intersect at a junction. Even then your very dependant on the
>>>>>> axial
>>>>>> lines all being slightly long and precisely how they intersect to
>>>>>> form lots of mini triangles.
>>>>>>
>>>>>> Believe me I programmed in clustering coefficient into
>>>>>> webmap@home
>>>>>> and didn't get anything exciting out of it. Basically clustering
>>>>>> coefficient works on the basis that If A knows B and B knows C
>>>>>> then
>>>>>> it is likely that C knows A. This is how all the social
>>>>>> networking
>>>>>> stuff works.
>>>>>>
>>>>>> For an axial map If street A connects to Street B and Street B
>>>>>> connects to Street C then it is highly unlikely that street C
>>>>>> connects to Street A (in fact the reverse is more true).
>>>>>>
>>>>>> Same for convex spaces but not for isovist grids.
>>>>>>
>>>>>> If you could go out to radius 3 or 4 the case would be
>>>>>> different but
>>>>>> this is not how Watts and Storogatz defined it just
>> degree/connectivity.
>>>>>>
>>>>>> Notice we are in an interesting twilight world where axial
>>>>>> maps are
>>>>>> largely 'scale free' (some highly connected hubs, most are low
>>>>>> connections) but not small world. This is a shame as if they
>>>>>> were we
>>>>>> could use the Derek J. de Solla Price generative mechanism
>>>>>> and be
>>>>>> able to run the growth of cities into the future.
>>>>>>
>>>>>> so short of redefining what you mean by small world axial maps
>>>>>> are
>>>>>> not small world and so moderately unique and so abnormal.
>>>>>>
>>>>>> sheep
>>>>>>
>>>>>
>>>> --
>>>> Course Director
>>>> MSc Adaptive Architecture & Computation
>>>> UCL Bartlett School of Graduate Studies
>>>>
>>>> http://www.vr.ucl.ac.uk/people/alasdair
>>>>
>>>
>>>
>>
>> --
>> Course Director
>> MSc Adaptive Architecture & Computation
>> UCL Bartlett School of Graduate Studies
>>
>> http://www.vr.ucl.ac.uk/people/alasdair
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